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%I M4613 N1968 #70 Jan 04 2025 06:19:17
%S 0,1,9,34,95,210,406,740,1161,1920,2695,4116,5369,7868,9690,13640,
%T 16116,22419,25365,34160,38640,50622,55154,73320,77225,100100,107730,
%U 135576,141085,182340,184760,233616,243408,297738,301420,385110,377511,467210,478842
%N a(n) = Sum_{k=1..n-1} k*sigma(k)*sigma(n-k).
%C Apart from initial zero this is the convolution of A340793 and A143128. - _Omar E. Pol_, Feb 16 2021
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D Jacques Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39.
%H Vaclav Kotesovec, <a href="/A000441/b000441.txt">Table of n, a(n) for n = 1..1000</a>
%H Jacques Touchard, <a href="/A000385/a000385.pdf">On prime numbers and perfect numbers</a>, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]
%F Convolution of A000203 with A064987. - _Sean A. Irvine_, Nov 14 2010
%F G.f.: x*f(x)*f'(x), where f(x) = Sum_{k>=1} k*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Apr 28 2018
%F a(n) = (n/24 - n^2/4)*sigma_1(n) + (5*n/24)*sigma_3(n). - _Ridouane Oudra_, Sep 17 2020
%F Sum_{k=1..n} a(k) ~ Pi^4 * n^5 / 2160. - _Vaclav Kotesovec_, May 09 2022
%p S:=(n,e)->add(k^e*sigma(k)*sigma(n-k),k=1..n-1);
%p f:=e->[seq(S(n,e),n=1..30)];f(1); # _N. J. A. Sloane_, Jul 03 2015
%t a[n_] := Sum[k*DivisorSigma[1, k]*DivisorSigma[1, n-k], {k, 1, n-1}]; Array[a, 40] (* _Jean-François Alcover_, Feb 08 2016 *)
%o (PARI) a(n) = sum(k=1, n-1, k*sigma(k)*sigma(n-k)); \\ _Michel Marcus_, Feb 02 2014
%o (PARI) a(n) = my(f = factor(n)); ((n - 6*n^2) * sigma(f) + 5*n * sigma(f, 3)) / 24; \\ _Amiram Eldar_, Jan 04 2025
%o (Python)
%o from sympy import divisor_sigma
%o def A000441(n): return (n*(1-6*n)*divisor_sigma(n)+5*n*divisor_sigma(n,3))//24 # _Chai Wah Wu_, Jul 25 2024
%Y Cf. A000385, A000477, A000499, A259692, A259693, A259694, A259695, A259696.
%Y Cf. A000203 (sigma_1), A001158 (sigma_3), A064987.
%Y Cf. A143128, A340793.
%K nonn,changed
%O 1,3
%A _N. J. A. Sloane_
%E More terms from _Sean A. Irvine_, Nov 14 2010
%E a(1)=0 prepended by _Michel Marcus_, Feb 02 2014